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(3x^3-4x^2+3x)/(x^2+1)dx
Similar expressions
(3x^3-4x^2+3x)/(x^2-1)
(3x^3+4x^2+3x)/(x^2+1)
(3x^3-4x^2-3x)/(x^2+1)

Solving integrals/ (3x^3-4x^2+3x)/(x^2+1)

Integral of (3x^3-4x^2+3x)/(x^2+1) dx

The solution

You have entered [src]

  1                     
  /                     
 |                      
 |     3      2         
 |  3*x  - 4*x  + 3*x   
 |  ----------------- dx
 |         2            
 |        x  + 1        
 |                      
/                       
0

$$\int\limits_{0}^{1} \frac{3 x^{3} - 4 x^{2} + 3 x}{x^{2} + 1}\, dx$$

Simplify

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{3 x^{3} - 4 x^{2} + 3 x}{x^{2} + 1} = 3 x - 4 + \frac{4}{x^{2} + 1}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 x\, dx = 3 \int x\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
    So, the result is: $\frac{3 x^{2}}{2}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(-4\right)\, dx = - 4 x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{4}{x^{2} + 1}\, dx = 4 \int \frac{1}{x^{2} + 1}\, dx$
    1. The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$ .
    So, the result is: $4 \operatorname{atan}{\left(x \right)}$
  The result is: $\frac{3 x^{2}}{2} - 4 x + 4 \operatorname{atan}{\left(x \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{3 x^{3} - 4 x^{2} + 3 x}{x^{2} + 1} = \frac{3 x^{3}}{x^{2} + 1} - \frac{4 x^{2}}{x^{2} + 1} + \frac{3 x}{x^{2} + 1}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{3 x^{3}}{x^{2} + 1}\, dx = 3 \int \frac{x^{3}}{x^{2} + 1}\, dx$
    1. Let $u = x^{2}$ .
      
      Then let $du = 2 x dx$ and substitute $du$ :
      
      $\int \frac{u}{2 u + 2}\, du$
      
      Rewrite the integrand:
      
      $\frac{u}{2 u + 2} = \frac{1}{2} - \frac{1}{2 \left(u + 1\right)}$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, du = \frac{u}{2}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{2 \left(u + 1\right)}\right)\, du = - \frac{\int \frac{1}{u + 1}\, du}{2}$
      
      Let $u = u + 1$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(u + 1 \right)}$
      
      So, the result is: $- \frac{\log{\left(u + 1 \right)}}{2}$
      
      The result is: $\frac{u}{2} - \frac{\log{\left(u + 1 \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{x^{2}}{2} - \frac{\log{\left(x^{2} + 1 \right)}}{2}$
    So, the result is: $\frac{3 x^{2}}{2} - \frac{3 \log{\left(x^{2} + 1 \right)}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{4 x^{2}}{x^{2} + 1}\right)\, dx = - 4 \int \frac{x^{2}}{x^{2} + 1}\, dx$
    1. Rewrite the integrand:
      
      $\frac{x^{2}}{x^{2} + 1} = 1 - \frac{1}{x^{2} + 1}$
    2. Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x^{2} + 1}\right)\, dx = - \int \frac{1}{x^{2} + 1}\, dx$
      
      The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$ .
      
      So, the result is: $- \operatorname{atan}{\left(x \right)}$
      
      The result is: $x - \operatorname{atan}{\left(x \right)}$
    So, the result is: $- 4 x + 4 \operatorname{atan}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{3 x}{x^{2} + 1}\, dx = 3 \int \frac{x}{x^{2} + 1}\, dx$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{x}{x^{2} + 1}\, dx = \frac{\int \frac{2 x}{x^{2} + 1}\, dx}{2}$
      
      Let $u = x^{2} + 1$ .
      
      Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{2 u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x^{2} + 1 \right)}$
      
      So, the result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2}$
    So, the result is: $\frac{3 \log{\left(x^{2} + 1 \right)}}{2}$
  The result is: $\frac{3 x^{2}}{2} - 4 x + 4 \operatorname{atan}{\left(x \right)}$
Add the constant of integration:

$\frac{3 x^{2}}{2} - 4 x + 4 \operatorname{atan}{\left(x \right)}+ \mathrm{constant}$

The answer is:

$\frac{3 x^{2}}{2} - 4 x + 4 \operatorname{atan}{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                 
 |                                                  
 |    3      2                                     2
 | 3*x  - 4*x  + 3*x                            3*x 
 | ----------------- dx = C - 4*x + 4*atan(x) + ----
 |        2                                      2  
 |       x  + 1                                     
 |                                                  
/

$$4\,\arctan x+{{3\,x^2-8\,x}\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-5/2 + pi

$${{2\,\pi-5}\over{2}}$$

-5/2 + pi

$$- \frac{5}{2} + \pi$$

Simplify

Numerical answer [src]

0.641592653589793

0.641592653589793

The graph

Use the examples entering the upper and lower limits of integration.

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Identical expressions

Similar expressions

Integral of (3x^3-4x^2+3x)/(x^2+1) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2